What is the integral of sin^2(mx) ?

The integral of sin^2(mx) is 1/2 x-1/(4m)sin(2mx)+C

integral of sin^2(mx)

\int sin^{2} (m x) d x

\frac{1}{2} x - \frac{1}{4 m} sin (2 m x) + C

Solution steps

\int sin^{2} (m x) d x

Rewrite using trig identities

= \int 1 - cos^{2} (m x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x - \int cos^{2} (m x) d x

\int 1 d x = x

\int cos^{2} (m x) d x = \frac{1}{2} (x + \frac{1}{2 m} sin (2 m x))

= x - \frac{1}{2} (x + \frac{1}{2 m} sin (2 m x))

Simplify

x - \frac{1}{2} (x + \frac{1}{2 m} sin (2 m x)) : \frac{1}{2} x - \frac{1}{4 m} sin (2 m x)

= \frac{1}{2} x - \frac{1}{4 m} sin (2 m x)

Add a constant to the solution

= \frac{1}{2} x - \frac{1}{4 m} sin (2 m x) + C

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What is the integral of sin^2(mx) ?
The integral of sin^2(mx) is 1/2 x-1/(4m)sin(2mx)+C